Answer
$a.\,\,\,p(A\cap B) =\left \{ \varnothing ,\left \{ 2 \right \} \right \}$
$b.\,\,p(A)=\left \{ \varnothing ,\left \{ 1 \right \},\left \{ 2 \right \},\left \{ 1,2 \right \} \right \} $
$c.\,\,p(A\cup B)=\left \{ \varnothing ,\left \{ 1 \right \},\left \{ 2 \right \} ,\left \{ 3 \right \},\left \{ 1,2 \right \},\left \{ 1,3 \right \},\left \{ 2,3 \right \},\left \{ 1,2,3 \right \}\right \} $
$d.\,\,$
$p(A\times B)=\left \{ \varnothing ,\left \{ (1,2) \right \},\left \{ (1,3) \right \},\left \{ (2,2) \right \},\left \{ (2,3) \right \},\left \{ (1,2),(1,3) \right \},\left \{ (1,2),(2,2) \right \},\left \{ (1,2),(2,3) \right \},\left \{ (1,3),(2,2) \right \},\left \{ (1,3),(2,3) \right \},\left \{ (2,2),(2,3) \right \},\left \{(1,2),(1,3),(2,2) \right \},\left \{ (1,2),(1,3),(2,3) \right \},\left \{ (1,2),(2,2),(2,3) \right \},\left \{ (1,3),(2,2),(2,3) \right \},\left \{ (1,2),(1,3),(2,2),(2,3) \right \} \right \}$
Work Step by Step
$A=\left \{ 1,2 \right \}\,\,B=\left \{ 2,3 \right \}\,\,$
$A\cap B=\left \{ 2 \right \} $
$a.\,\,\,p(A\cap B) =\left \{ \varnothing ,\left \{ 2 \right \} \right \}$
$b.\,\,p(A)=\left \{ \varnothing ,\left \{ 1 \right \},\left \{ 2 \right \},\left \{ 1,2 \right \} \right \} $
$c.\,\,A\cup B=\left \{ 1,2,3 \right \} $
$so\,\,p(A\cup B)=\left \{ \varnothing ,\left \{ 1 \right \},\left \{ 2 \right \} ,\left \{ 3 \right \},\left \{ 1,2 \right \},\left \{ 1,3 \right \},\left \{ 2,3 \right \},\left \{ 1,2,3 \right \}\right \} $
$d.\,\,A\times B=\left \{ (1,2),(1,3),(2,2),(2,3) \right \}$
$so\,\,$
$p(A\times B)=\left \{ \varnothing ,\left \{ (1,2) \right \},\left \{ (1,3) \right \},\left \{ (2,2) \right \},\left \{ (2,3) \right \},\left \{ (1,2),(1,3) \right \},\left \{ (1,2),(2,2) \right \},\left \{ (1,2),(2,3) \right \},\left \{ (1,3),(2,2) \right \},\left \{ (1,3),(2,3) \right \},\left \{ (2,2),(2,3) \right \},\left \{(1,2),(1,3),(2,2) \right \},\left \{ (1,2),(1,3),(2,3) \right \},\left \{ (1,2),(2,2),(2,3) \right \},\left \{ (1,3),(2,2),(2,3) \right \},\left \{ (1,2),(1,3),(2,2),(2,3) \right \} \right \}$
